### A Pluto.jl notebook ###
# v0.18.4

using Markdown
using InteractiveUtils

# ╔═╡ ae47661e-95a2-11ec-119b-8194fe27f96e
begin
	using Plots
	using LsqFit
	import Distributions as di
	using Random
	using Turing
	using Statistics
	using StatsPlots
	using FillArrays
end

# ╔═╡ 8dc3dee2-a7d2-4afd-94b1-63154910a54c
t = 0:0.1:5

# ╔═╡ 4c8163a1-1206-47dc-bc58-1d642663a22e
function model(t,p)
	return p[1] .* exp.(-t/p[2])
end

# ╔═╡ 38284326-0524-43cd-8fbe-3d228aac9bf4
begin
	data_err = 0.5
	d = di.Normal(0,data_err)
end

# ╔═╡ a0add249-405b-4f30-a667-c791d155d5ca
ampl_range = 2:0.1:5

# ╔═╡ c5f1d281-fc40-4fdb-ae25-664ff7c99332
decay_range = 1:0.1:3

# ╔═╡ 2288748b-9ae9-4032-a371-b2de8712cd88
y = 3.0 .* exp.(-t/2.0) + rand(d,length(t))

# ╔═╡ 7abe92c4-3081-429d-88b0-fd1d367fb3df
begin
	plot(t,y,yerror=data_err)
	plot!(t,model(t,[3.0,2.0]))
end

# ╔═╡ 8948ee55-fa92-4ac5-9836-ff2640a541db
fit = curve_fit(model, t, y, [1.0,1.0])

# ╔═╡ eb879e89-7277-4804-952b-19048874ced5
fit.param

# ╔═╡ 3a803151-2291-4798-b1f4-15e6bd1189fb
cov = estimate_covar(fit)

# ╔═╡ 7cbfb55e-8984-4bee-9848-dcaacdfb1282
conf = confidence_interval(fit, 0.1)

# ╔═╡ bf29e446-1cd4-4c2d-8f1f-0b91f39aa450
exp_dist = di.MvNormal(fit.param,cov)

# ╔═╡ 3449f76f-9dbe-4c07-898e-5b944174b374
function f(x,y)
	return di.pdf(exp_dist,[x,y])
end

# ╔═╡ d977a512-3085-4a87-a95d-7d3041402e0a
contourf(ampl_range,decay_range,f)

# ╔═╡ fc76af35-ac05-418a-993d-a382c41b036f
begin
	my_normal = di.Normal(1,2)
	my_sample = rand(my_normal,200)
end

# ╔═╡ 2dacbdf4-dbf2-42c9-9529-007fff009c12
mean(my_sample)

# ╔═╡ 1d360be7-43a1-4a75-8d09-e229917ff815
std(my_sample)

# ╔═╡ 923e8ca4-8f66-47b8-a2b1-b7b384416163
@model function my_fit(data)
	μ ~ Uniform(-10,10)
	σ ~ Uniform(0,20)
	for i in 1:length(data)
		data[i] ~ Normal(μ,σ)
	end
end

# ╔═╡ b2c73bae-7a71-49b6-8ea0-3c57dd7843ae
model1 = my_fit(my_sample)

# ╔═╡ 46e1c786-6abc-4f82-a1a8-04e0394aa005
chain = sample(model1,NUTS(0.65),1000)

# ╔═╡ 39dc3dae-105b-490c-a5af-b8cfdc059bc0
StatsPlots.plot(chain)

# ╔═╡ 7241b49c-b5d9-438a-950a-8ffcfa3d301b
describe(chain)

# ╔═╡ c17062e7-42dc-4b49-8b54-2a61adf6396b
x = 0:10

# ╔═╡ fee857c0-cbb4-4a49-ab9e-59c58b92cb63
function lin_model(x,p)
	return p[1] .+ (p[2] .* x)
end

# ╔═╡ 7bb4d1f9-bfa7-4aef-b88e-96cb6bef96ee
function quad_model(x,p)
	return p[1] .+ (p[2] .* x) .+ (p[3] .* x .^ 2)
end

# ╔═╡ 974540df-328a-4855-a954-028c19f1ea56
begin
	dlin_error = 2.0
	dlin = di.Normal(0,dlin_error)
end

# ╔═╡ 029632a7-e923-448b-86b9-21e5c9c6acd6
yy = 2.0 .+ (3.0 .* x) .+ 0.1 .* x .^ 2 .+ rand(dlin,length(x))

# ╔═╡ a56ca084-6366-4148-b600-56f797d3927e
fit_lin = curve_fit(lin_model,x,yy,[2.0,3.0])

# ╔═╡ 57f154a5-d32c-47f0-9dec-a136b3fd44a7
fit_lin.param

# ╔═╡ 89f26e4b-f10e-47e2-a032-b3848ec618f2
yy_fit = fit_lin.param[1] .+ fit_lin.param[2] .* x

# ╔═╡ 07174e20-8292-441b-ae92-51800f641e64
yy_quadfit = curve_fit(quad_model, x,yy,[3.0,2.0,0.0])

# ╔═╡ 4b4cf4ee-27b8-4d25-a926-0c4e311c980f
yy_quadfit.param

# ╔═╡ 08e5fe39-a598-4f06-9c5c-aa8e7ac3c4b9
yy_quad = yy_quadfit.param[1] .+ (yy_quadfit.param[2] .* x) .+ (yy_quadfit.param[3] .* x .^ 2)

# ╔═╡ 89a7fbc5-ea56-4e8d-b50d-63456484dca0
cov_quad = estimate_covar(yy_quadfit)

# ╔═╡ 461b0327-1278-4c99-90f9-b8c1b6e1a5c6
delta_quad = sqrt(cov_quad[3,3])

# ╔═╡ afb2e5ef-3d4c-439a-8574-443a09eb850d
begin
	scatter(x,yy,yerror=dlin_error,label=false)
	plot!(x, yy_fit, label=false)
	plot!(x, yy_quad, label=false)
end

# ╔═╡ 0246ae6c-974f-4e4e-a16f-6dce6a0cddfa
chi_quad = sum(yy_quadfit.resid .^ 2 /dlin_error^2)

# ╔═╡ 838f4345-c865-4712-8d39-b52777ab986d
chi_lin = sum(fit_lin.resid .^ 2 /dlin_error^2)

# ╔═╡ 44503a57-a47c-46dd-9882-c97c568b4bc7
exp(-chi_lin)/exp(-chi_quad)*1.0/sqrt(cov_quad[3,3])/sqrt(2*π)

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
FillArrays = "1a297f60-69ca-5386-bcde-b61e274b549b"
LsqFit = "2fda8390-95c7-5789-9bda-21331edee243"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
Turing = "fce5fe82-541a-59a6-adf8-730c64b5f9a0"

[compat]
Distributions = "~0.25.53"
FillArrays = "~0.13.2"
LsqFit = "~0.12.1"
Plots = "~1.27.5"
StatsPlots = "~0.14.33"
Turing = "~0.21.1"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

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[[TableTraits]]
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git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "OrderedCollections", "TableTraits", "Test"]
git-tree-sha1 = "5ce79ce186cc678bbb5c5681ca3379d1ddae11a1"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.7.0"

[[Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[TerminalLoggers]]
deps = ["LeftChildRightSiblingTrees", "Logging", "Markdown", "Printf", "ProgressLogging", "UUIDs"]
git-tree-sha1 = "62846a48a6cd70e63aa29944b8c4ef704360d72f"
uuid = "5d786b92-1e48-4d6f-9151-6b4477ca9bed"
version = "0.1.5"

[[Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[Tracker]]
deps = ["Adapt", "DiffRules", "ForwardDiff", "LinearAlgebra", "LogExpFunctions", "MacroTools", "NNlib", "NaNMath", "Printf", "Random", "Requires", "SpecialFunctions", "Statistics"]
git-tree-sha1 = "0874c1b5de1b5529b776cfeca3ec0acfada97b1b"
uuid = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c"
version = "0.2.20"

[[Transducers]]
deps = ["Adapt", "ArgCheck", "BangBang", "Baselet", "CompositionsBase", "DefineSingletons", "Distributed", "InitialValues", "Logging", "Markdown", "MicroCollections", "Requires", "Setfield", "SplittablesBase", "Tables"]
git-tree-sha1 = "c76399a3bbe6f5a88faa33c8f8a65aa631d95013"
uuid = "28d57a85-8fef-5791-bfe6-a80928e7c999"
version = "0.4.73"

[[TreeViews]]
deps = ["Test"]
git-tree-sha1 = "8d0d7a3fe2f30d6a7f833a5f19f7c7a5b396eae6"
uuid = "a2a6695c-b41b-5b7d-aed9-dbfdeacea5d7"
version = "0.3.0"

[[Turing]]
deps = ["AbstractMCMC", "AdvancedHMC", "AdvancedMH", "AdvancedPS", "AdvancedVI", "BangBang", "Bijectors", "DataStructures", "Distributions", "DistributionsAD", "DocStringExtensions", "DynamicPPL", "EllipticalSliceSampling", "ForwardDiff", "Libtask", "LinearAlgebra", "MCMCChains", "NamedArrays", "Printf", "Random", "Reexport", "Requires", "SciMLBase", "SpecialFunctions", "Statistics", "StatsBase", "StatsFuns", "Tracker", "ZygoteRules"]
git-tree-sha1 = "ef0fdc72023c4480a9372f32db88cce68b186e8a"
uuid = "fce5fe82-541a-59a6-adf8-730c64b5f9a0"
version = "0.21.1"

[[URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
version = "1.0.2"

[[Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[Unzip]]
git-tree-sha1 = "34db80951901073501137bdbc3d5a8e7bbd06670"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.1.2"

[[Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[Widgets]]
deps = ["Colors", "Dates", "Observables", "OrderedCollections"]
git-tree-sha1 = "505c31f585405fc375d99d02588f6ceaba791241"
uuid = "cc8bc4a8-27d6-5769-a93b-9d913e69aa62"
version = "0.6.5"

[[WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "de67fa59e33ad156a590055375a30b23c40299d3"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "0.5.5"

[[XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
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# ╠═974540df-328a-4855-a954-028c19f1ea56
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002


"""
    initialScenario(c, d, Γ)

Returns a good initial scenario ``c_0``. It is used in computation of [`evaluationProblem`](@ref)
and [`adversarialProblem`](@ref).

Check section 5.1 _Adversarial lower bound_ of [publication](https://arxiv.org/abs/1811.06719)
for more information about this algorithm.

# Arguments
- `c`: vector of nonnegative nominal second stage costs.
- `d`: vector of maximal deviations of the costs from their nominal values.
- `Γ`: budget, or the amount of uncertainty, which can be allocated to the second stage costs
"""
function initialScenario(c, d, Γ)
    calculateDelta(c, d, m) = collect(max(min(d[i], m - c[i]), 0) for i in 1:length(c))

    ϵ = 0.001
    l = 0
    m = 0
    r = maximum(map((t) -> sum(t), zip(c, d)))

    δ = []
    while abs(l - r) >= ϵ
        m = (l + r) / 2.0
        δ = calculateDelta(c, d, m)
        if sum(δ) > Γ
            r = m
        else
            l = m
        end
    end

    δ = reshape(calculateDelta(c, d, m), size(c))
    return c + δ
end


function checkSVDDone(S::Vector, thresh::Float64)
  N = length(S)
  (N <= 1 || thresh < 0.0) && return (true, 1)
  S1t = S[1] * thresh
  start = 2
  while start <= N
    (S[start] < S1t) && break
    start += 1
  end
  if start >= N
    return (true, N)
  end
  return (false, start)
end

function svd_recursive(M::AbstractMatrix; thresh::Float64=1E-3, north_pass::Int=2)
  Mr, Mc = size(M)
  if Mr > Mc
    V, S, U = svd_recursive(transpose(M))
    conj!(U)
    conj!(V)
    return U, S, V
  end

  #rho = BLAS.gemm('N','T',-1.0,M,M) #negative to sort eigenvalues greatest to smallest
  rho = -M * M' #negative to sort eigenvalues in decreasing order
  D, U = eigen(Hermitian(rho), 1:size(rho, 1))

  Nd = length(D)

  V = M' * U

  V, R = qr_positive(V)
  for n in 1:Nd
    D[n] = R[n, n]
  end

  (done, start) = checkSVDDone(D, thresh)

  done && return U, D, V

  u = view(U, :, start:Nd)
  v = view(V, :, start:Nd)

  b = u' * (M * v)
  bu, bd, bv = svd_recursive(b; thresh=thresh, north_pass=north_pass)

  u .= u * bu
  v .= v * bv
  view(D, start:Nd) .= bd

  return U, D, V
end

# TODO: maybe move to another location?
# Include options for other svd algorithms
function polar(M::AbstractMatrix)
  U, S, V = svd(M) # calls LinearAlgebra.svd(_)
  return U * V', V * Diagonal(S) * V'
end


@with_kw struct FastTree
    tolerance::Float64 = 1.0
end

# This is the main function

function solve(solver::FastTree, problem::Problem)
    center = problem.input.center
    radius = problem.input.radius[1]
    (W, b) = (problem.network.layers[1].weights, problem.network.layers[1].bias)

    input = forward_affine_map(solver, W, b, problem.input)
    lower, upper = low(input), high(input)
    local_lower, local_upper = similar(lower), similar(lower)

    k_1 = size(W, 1)

    C = vcat(W, -W)
    d = zeros(2k_1)

    stack = Vector{Hyperrectangle}(undef, 0)
    push!(stack, input)
    #count = 0
    while !isempty(stack)
        interval = popfirst!(stack)
        reach = forward_network(solver, problem.network, interval)
        #count += 1
        if issubset(reach, problem.output)
            continue
        else
            if get_largest_width(interval) > solver.tolerance
                sections = bisect(interval)
                for i in 1:2
                    local_lower = low(sections[i])
                    local_upper = high(sections[i])
                    d = vcat(local_upper - b, b - local_lower)

                    inter = intersection(problem.input, HPolyhedron(C, d))

                    if isempty(inter) == false
                        push!(stack, sections[i])
                    end
                end
            else
                return BasicResult(:violated)
            end
        end
    end
    #println("DimTree($(solver.tolerance)): $(count)")
    return BasicResult(:holds)
end

function forward_network(solver::FastTree, nnet::Network, input::Hyperrectangle)
    layers = nnet.layers
    act = layers[1].activation
    reach = Hyperrectangle(low = act.(low(input)), high = act.(high(input)))

    for i in 2:length(layers)
        reach = forward_layer(solver, layers[i], reach)
    end
    return reach
end

function forward_layer(solver::FastTree, L::Layer, input::Hyperrectangle)
    (W, b, act) = (L.weights, L.bias, L.activation)
    center = zeros(size(W, 1))
    gamma  = zeros(size(W, 1))
    for j in 1:size(W, 1)
        node = Node(W[j,:], b[j], act)
        center[j], gamma[j] = forward_node(solver, node, input)
    end
    return Hyperrectangle(center, gamma)
end

function forward_node(solver::FastTree, node::Node, input::Hyperrectangle)
    output    = node.w' * input.center + node.b
    deviation = sum(abs.(node.w) .* input.radius)
    βmax = node.act(output + deviation)
    βmin = node.act(output - deviation)
    return ((βmax + βmin)/2, (βmax - βmin)/2)
end

function forward_affine_map(solver::FastTree, W::Matrix, b::Vector, input::Hyperrectangle)
    center = W * input.center + b
    radius = abs.(W) * input.radius
    return Hyperrectangle(center, radius)
end


using Unitful
using PlutoUI
using PhysicalConstants.CODATA2018
using Plots
using Printf

k_B = BoltzmannConstant
c_0 = SpeedOfLightInVacuum
h = PlanckConstant

function planck_spectrum(lambdas, T)
	res = []
	for λ in lambdas
		push!(
			res,
			ustrip(
				uconvert(
					u"W/m^3", 2*pi*h*c_0^2 / λ.^5 / (exp(h*c_0/λ/k_B/T) - 1))
				)
			)
	end
	return res
end

function add_visible_light(lambda_unit, height, alpha)
	bar!(
		[ustrip(uconvert(lambda_unit, 400u"nm"))],
		[height],
		fillcolor=[:violet],
		bar_width=ustrip(uconvert(lambda_unit, 40u"nm")),
		fillalpha=alpha,
		label="",
	)
	bar!(
		[ustrip(uconvert(lambda_unit, 455u"nm"))],
		[height],
		fillcolor=[:blue],
		bar_width=ustrip(uconvert(lambda_unit, 70u"nm")),
		fillalpha=alpha,
		label="",
	)
	bar!(
		[ustrip(uconvert(lambda_unit, 532.5u"nm"))],
		[height],
		fillcolor=[:green],
		bar_width=ustrip(uconvert(lambda_unit, 85u"nm")),
		fillalpha=alpha,
		label="",
	)
	bar!(
		[ustrip(uconvert(lambda_unit, 580u"nm"))],
		[height],
		fillcolor=[:yellow],
		bar_width=ustrip(uconvert(lambda_unit, 10u"nm")),
		fillalpha=alpha,
		label="",
	)
	bar!(
		[ustrip(uconvert(lambda_unit, 617.5u"nm"))],
		[height],
		fillcolor=[:orange],
		bar_width=ustrip(uconvert(lambda_unit, 65u"nm")),
		fillalpha=alpha,
		label="",
	)
	bar!(
		[ustrip(uconvert(lambda_unit, 700u"nm"))],
		[height],
		fillcolor=[:red],
		bar_width=ustrip(uconvert(lambda_unit, 100u"nm")),
		fillalpha=alpha,
		label="",
	)
end

function plot_stuff(T)
    ls3 = 10 .^ (range(-2, stop=4, length=1000))
    y = planck_spectrum(ls3.*1u"nm", T*1u"K")
    add_visible_light(u"nm", maximum(y), 0.7)
    plot!(
        ls3,
        #ls3,
        y,
        #xscale=:log10,
        #yscale=:log10,
        xlim=[100, 1500],
        xlabel="Wellenlänge [nm]",
        ylabel="Intensität [W/m³]",
        label="$T K",
        color=:black,
        grid=true,
        lw=3
    )
end

for T in [3000, 6000, 12000]
    pgfplotsx()
    pl = plot()
    plot_stuff(T)
    savefig(pl, "planck_$T.tex")

    gr()
    pl = plot()
    plot_stuff(T)
    savefig(pl, "planck_$T.png")

    pl = plot()
    plot_stuff(T)
    savefig(pl, "planck_$T.svg")
end


using FractionalDiffEq, Plots, LaTeXStrings

s="\$D^{0.5}y(x)=1-y,\\ y(0)=0\$"

fun(x, y) = 1-y
prob=FDEProblem(fun, 0.5, 0, 5, 0.01)
result=solve(prob, PECE())
tspan=collect(0:0.01:5)

plot(tspan, result, title=s, linewidth=2, legend=:bottomright)


using Distributions

p = 6
λ = 9

function K(t)
    return 2λ*t / (1-2t) - p / 2 * log(1-2t)
end

function D1K(t)
    return ( 4λ*t ) / (1-2t)^2 + ( 2λ + p ) / ( 1 - 2t )
end

function D2K(t)
    return 2(p*(1-2t) + 4λ) / (1-2t)^3
end

function logf(n, t)
    return n * (K(t) - t*D1K(t)) + 0.5log(D2K(t))
end

function logg(n, t)
    return -1n * D2K(0) * t^2/2
end

function mh_saddle(T::Int = 10000; n::Int = 1)
    Z = zeros(T)
    g = Normal(0, 1 / sqrt(n*D2K(0)))
    for t = 1:T-1
        z = rand(g)
        logr = logf(n, z) - logf(n, Z[t]) + logg(n, Z[t]) - logg(n, z)
        if log(rand()) < logr
            Z[t+1] = z
        else
            Z[t+1] = Z[t]
        end
    end
    return Z
end

Z = mh_saddle(n = 1)

function tau(x)
    return ( -1p + 2x - sqrt(p^2 + 8λ * x) ) / (4x)
end

println(sum(Z .> tau(36.225)) / 10000)
println(sum(Z .> tau(40.542)) / 10000)
println(sum(Z .> tau(49.333)) / 10000)

# n = 10
Z = mh_saddle(n = 10)

println(sum(Z .> tau(113.6667)) / 10000)
println(sum(Z .> tau(102.063)) / 10000)
println(sum(Z .> tau(96.19335)) / 10000)

# n = 100
Z = mh_saddle(n = 100)

println(sum(Z .> tau(25.18054)) / 10000)
println(sum(Z .> tau(25.52361)) / 10000)
println(sum(Z .> tau(26.17395)) / 10000)


# example showing how to do maximum likelihood estimation
# for data generated by CLRM with normality
# sample size
using Econometrics, Statistics
n = 10

# random true parameters
theta = [1.0,-1.0,0.0,1.0]
b = theta[1:3]
sig = theta[4]

# generate random data
x = [ones(n,1) rand(n,2)]
e = sig*randn(n,1)
y = x*b + e

# do ML: note minus sign, also, do "edit(normal)" to see what's done
obj = theta -> -mean(normal(theta, y, x))
thetahat, junk, junk = fminunc(obj, theta)

# results
println("the true parameters: ", theta)
println("the ML estimates: ", thetahat)
println("the OLS estimates: ", inv(x'*x)*x'*y)


using Plots

# hyperbola.png --------------------------------------------------------------------------

paramlen=1.5
t0=0.6*paramlen
ran = LinRange(-paramlen, paramlen, 100)

# Hyperbola
plot(framestyle=:origin, aspect_ratio=:equal)
plot!(cosh, sinh, ran, color=:black, lab="")
plot!(t -> -cosh(t), sinh, ran, color=:black, lab="")

# Guides
plot!(t -> t, t -> t, -sinh(last(ran)):0.01:sinh(last(ran)), style=:dash, color=:red, lab="")
plot!(t -> t, t -> -t, -sinh(last(ran)):0.01:sinh(last(ran)), style=:dash, color=:red, lab="")
plot!([(0, sinh(t0)), (cosh(t0), sinh(t0))], style=:dot, color=:black, lab="")
plot!([(cosh(t0), 0), (cosh(t0), sinh(t0))], style=:dot, color=:black, lab="")

# Annotations
annotate!(cosh(t0)+0.1, sinh(t0)/2, Plots.text("sinh t", :left, 8))
annotate!(cosh(t0)/2, sinh(t0)+0.1, Plots.text("cosh t", :up, 8))
savefig("hyperbola.png")

# hyperbolic-paraboloid.png -------------------------------------------------------------

x = LinRange(-10, 10, 100)
y = LinRange(-10, 10, 100)
z(x, y) = x^2-y^2
plot(x, y, z, st=:surface)
savefig("hyperbolic-paraboloid.png")

# image-parametrization-Y.png ------------------------------------------------------------

x = LinRange(-10, 10, 100)
y = LinRange(-10, 10, 100)
z(x, y) = abs(y) <= abs(x) ? x^2-y^2 : NaN
plot(x, y, z, st=:surface)
savefig("image-parametrization-Y.png")


import MultipleShootingPEM
ms = MultipleShootingPEM
using Plots
pyplot()
using LaTeXStrings

# Generate Data
k0 = 1
N = 200
θ_opt = 3.78
function logistic_f(x_next, x, k, θ)
    x_next .=  θ*x - θ*x.^2
    return
end
function logistic_g(y, x, k, θ)
    y .= x
    return
end
# Define initial conditions
x0 = [0.5]
y = Vector{Vector{Float64}}(N)
for i = 1:length(y)
    y[i] = zeros(1)
end
x_final = ms.simulate_space_state!(y, logistic_f, logistic_g,
                                   x0, (k0, k0+N-1), (θ_opt,))

# Multiple shooting error estimation
M = 40
k0_list = collect(k0:Int(N//M):k0+N-1)
x0_list = y[k0_list]
list_procs = ones(Int, M)

# Define polynomial model
# f(x) = θ1*x - θ1*x^2
@everywhere function f(y, dx, dθ, x, k, θ)
    y .= θ[1]*x - θ[1]*x.^2
    dx .= θ[1] - 2*θ[1]*x
    dθ .=  x-x.^2
    return
end
# g(x) = x
@everywhere function g(y, dx, dθ, x, k, θ)
    y .= x
    dx .= 1
    dθ .= 0
    return
end

θ = 0
multiple_shoot = ms.MultipleShooting(f, g, x0_list,
                                     y, k0_list, θ,
                                     list_procs)
ms.cost_function(multiple_shoot)

# Get cost function
θ_min = 2.0
θ_delta = 0.01
θ_max = 4.0
θ_range = θ_min:θ_delta:θ_max
npoints = length(θ_range)
cost = Vector{Float64}(npoints)
i = 1
for θ = θ_range
    ms.new_simulation!(multiple_shoot, x0_list, [θ])
    cost[i] = ms.cost_function(multiple_shoot)
    i += 1
end

nrealiz = 20
cost_matrix = Matrix{Float64}(npoints, nrealiz)
srand(1)
j = 1
for j = 1:nrealiz
    σ = 0.005
    x0_list_modified = copy(x0_list)
    for i = 1:M
        x0_list_modified .+= σ*randn()
    end
    i = 1
    for θ = θ_range
        ms.new_simulation!(multiple_shoot, x0_list_modified, [θ])
        cost_matrix[i, j] = ms.cost_function(multiple_shoot)
        i += 1
    end
    j += 1
end

# Get solver solution
srand(4)
θ_list = Float64[]
cost_list = Float64[]
res_list = []
for θp = 3.2:0.05:3.9
    σ = 0.002
    x0_list_modified = copy(x0_list)
    for i = 1:M
        x0_list_modified .+= σ*randn()
    end
    θ0 = [θp]
    opt = ms.OptimizationProblem(f, g, x0_list_modified, y,
                                 k0_list, θ0, list_procs)

    res = ms.solve(opt, options=Dict("gtol" => 1e-10,
                                     "xtol" => 1e-10,
                                     "maxiter" => 2000))
    θ_est = res["x"][1]
    cost_est = res["fun"]

    push!(θ_list, θ_est)
    push!(cost_list, cost_est)
    push!(res_list, res)
end

plot(θ_range, cost, color=:black, lw=2, legend=false, grid=false,
     xlims=[θ_min, θ_max], xguide=L"\theta", yguide=L"V(\theta)")
plot!(θ_range, cost_matrix, color=:blue, linealpha=0.2)
scatter!(θ_list, cost_list, marker=:c, color=:green, markersize=5)
vline!([θ_opt], ls=:dot, color=:red)

# savefig("ratio4_ms_logistic.tex")

# Evaluate solver performance
niter_list = [res["niter"] for res in res_list]
median(niter_list)
maximum(niter_list)
minimum(niter_list)

nfun_list = [res["nfev"] for res in res_list]
median(nfun_list)
maximum(nfun_list)
minimum(nfun_list)

exect_list = [res["execution_time"] for res in res_list]
median(exect_list)
maximum(exect_list)
minimum(exect_list)


# SEIR Model

using DifferentialEquations

beta = 0.215; gamma = 0.07; N = 7000;
alpha = 0.25;

"""SEIR Model
Extension of the basic SIR model

"""
function SEIR!(du,u,p,t)
    du[1] = -beta* u[1]/N*u[3]
    du[2] = beta*u[1]/N*u[3] - alpha*u[2]
    du[3] = alpha* u[2] - gamma*u[3]
    du[4] = gamma*u[3]
end

   u0 = [N-15.0;15.0;0.0;0.0]
   tspan = (0.0,100.0)
   prob = ODEProblem(SEIR!,u0,tspan)
   sol = solve(prob)

   plot(sol)


function betaBinomial(prior::P, bernoulliTrials::Vector{Int}) where {P <: Beta}
  n = length(bernoulliTrials)
  successes = sum(bernoulliTrials)
  failures = n - successes
  a = prior.α + successes
  b = prior.β + failures
  return Beta(a, b)
end
#betaBinomial(Beta(2,3), [1,0,1,1])


using LinearAlgebra

A = [1 1 -1; -2 0 1; 0 2 1]
B = inv(A)
E = B*A

println("A^{-1}=", B)
println("A*A^{-1}=", E)


function f753(N::Int, K = 0, n₃ = 0, n₅ = 0, n₇ = 0)
    if K > N
        return 0
    else
        result = (n₃ > 0 && n₅ > 0 && n₇ > 0) +
            f753(N, 10K + 3, n₃ + 1, n₅, n₇) +
            f753(N, 10K + 5, n₃, n₅ + 1, n₇) +
            f753(N, 10K + 7, n₃, n₅, n₇ + 1)
        return result
    end
end

function main()
    N = parse(Int, readline())
    println(f753(N))
end

main()


function adjacency_matrix(adj::AbstractMatrix, T::DataType=eltype(adj))
    m, n = size(adj)
    (m == n) || throw(DimensionMismatch("adjacency matrix is not a square matrix: ($m, $n)"))
    T.(adj)
end

function degrees(adj::AbstractMatrix; dir::Symbol=:out)
    if issymmetric(adj)
        d = vec(sum(adj, dims=1))
    else
        if dir == :out
            d = vec(sum(adj, dims=1))
        elseif dir == :in
            d = vec(sum(adj, dims=2))
        elseif dir == :both
            d = vec(sum(adj, dims=1)) + vec(sum(adj, dims=2))
        else
            throw(DomainError(dir, "invalid argument, only accept :in, :out and :both"))
        end
    end
    d
end

degrees(adj::AbstractMatrix, T::DataType; dir::Symbol=:out) = degrees(T.(adj); dir=dir)

"""
    degree_matrix(g[, T]; dir=:out)

Degree matrix of graph `g`. Return a matrix which contains degrees of each vertex in its diagonal.
The values other than diagonal are zeros.

# Arguments

- `g`: should be a adjacency matrix, `FeaturedGraph`, `SimpleGraph`, `SimpleDiGraph` (from Graphs)
    or `SimpleWeightedGraph`, `SimpleWeightedDiGraph` (from SimpleWeightedGraphs).
- `T`: result element type of degree vector; default is the element type of `g` (optional).
- `dir`: direction of degree; should be `:in`, `:out`, or `:both` (optional).

# Examples

```jldoctest
julia> using GraphLaplacians

julia> m = [0 1 1; 1 0 0; 1 0 0];

julia> GraphLaplacians.degree_matrix(m)
3×3 LinearAlgebra.Diagonal{Int64, Vector{Int64}}:
 2  ⋅  ⋅
 ⋅  1  ⋅
 ⋅  ⋅  1
```
"""
function degree_matrix(adj::AbstractMatrix, T::DataType=eltype(adj);
                       dir::Symbol=:out, squared::Bool=false, inverse::Bool=false)
    d = degrees(adj, T, dir=dir)
    squared && (d .= sqrt.(d))
    inverse && (d .= inv.(d); replace!(d, typemax(T)=>zero(T)))
    return Diagonal(T.(d))
end

"""
    laplacian_matrix(g[, T]; dir=:out)

Laplacian matrix of graph `g`.

# Arguments

- `g`: should be a adjacency matrix, `FeaturedGraph`, `SimpleGraph`, `SimpleDiGraph` (from Graphs)
    or `SimpleWeightedGraph`, `SimpleWeightedDiGraph` (from SimpleWeightedGraphs).
- `T`: result element type of degree vector; default is the element type of `g` (optional).
- `dir`: direction of degree; should be `:in`, `:out`, or `:both` (optional).
"""
Graphs.laplacian_matrix(adj::AbstractMatrix, T::DataType=eltype(adj); dir::Symbol=:out) =
    degree_matrix(adj, T, dir=dir) - SparseMatrixCSC(T.(adj))

"""
    normalized_laplacian(g[, T]; dir=:both, selfloop=false)

Normalized Laplacian matrix of graph `g`.

# Arguments

- `g`: should be a adjacency matrix, `FeaturedGraph`, `SimpleGraph`, `SimpleDiGraph` (from Graphs)
    or `SimpleWeightedGraph`, `SimpleWeightedDiGraph` (from SimpleWeightedGraphs).
- `T`: result element type of degree vector; default is the element type of `g` (optional).
- `selfloop`: adding self loop while calculating the matrix (optional).
- `dir`: direction of graph; should be `:in` or `:out` (optional).
"""
function normalized_laplacian(adj::AbstractMatrix, T::DataType=eltype(adj);
                              dir::Symbol=:both, selfloop::Bool=false)
    if dir == :both
        selfloop && (adj += I)
        inv_sqrtD = degree_matrix(adj, T, dir=:both, squared=true, inverse=true)
        return T.(I - inv_sqrtD * adj * inv_sqrtD)
    else
        return T.(I - degree_matrix(adj, T, dir=dir, inverse=true) * adj)
    end
end

@doc raw"""
    scaled_laplacian(g[, T])

Scaled Laplacien matrix of graph `g`,
defined as ``\hat{L} = \frac{2}{\lambda_{max}} L - I`` where ``L`` is the normalized Laplacian matrix.

# Arguments

- `g`: should be a adjacency matrix, `FeaturedGraph`, `SimpleGraph`, `SimpleDiGraph` (from Graphs)
    or `SimpleWeightedGraph`, `SimpleWeightedDiGraph` (from SimpleWeightedGraphs).
- `T`: result element type of degree vector; default is the element type of `g` (optional).
"""
function scaled_laplacian(adj::AbstractMatrix, T::DataType=eltype(adj))
    @assert issymmetric(adj) "scaled_laplacian only works with symmetric matrices"
    E = eigen(Symmetric(Array(adj))).values
    T(2. / maximum(E)) * normalized_laplacian(adj, T) - I
end

"""
    random_walk_laplacian(g[, T]; dir=:out)

Random walk normalized Laplacian matrix of graph `g`.

# Arguments

- `g`: should be a adjacency matrix, `FeaturedGraph`, `SimpleGraph`, `SimpleDiGraph` (from Graphs)
    or `SimpleWeightedGraph`, `SimpleWeightedDiGraph` (from SimpleWeightedGraphs).
- `T`: result element type of degree vector; default is the element type of `g` (optional).
- `dir`: direction of degree; should be `:in`, `:out`, or `:both` (optional).
"""
function random_walk_laplacian(adj::AbstractMatrix, T::DataType=eltype(adj); dir::Symbol=:out)
    d = degrees(adj, dir=dir)
    inv_d = 1 ./ d
    replace!(inv_d, typemax(float(T)) => zero(float(T)))  # avoid degree to be zero
    P = Diagonal(inv_d) * adj
    SparseMatrixCSC(T.(I - P))
end

"""
    signless_laplacian(g[, T]; dir=:out)

Signless Laplacian matrix of graph `g`.

# Arguments

- `g`: should be a adjacency matrix, `FeaturedGraph`, `SimpleGraph`, `SimpleDiGraph` (from Graphs)
    or `SimpleWeightedGraph`, `SimpleWeightedDiGraph` (from SimpleWeightedGraphs).
- `T`: result element type of degree vector; default is the element type of `g` (optional).
- `dir`: direction of degree; should be `:in`, `:out`, or `:both` (optional).
"""
signless_laplacian(adj::AbstractMatrix, T::DataType=eltype(adj); dir::Symbol=:out) =
    degree_matrix(adj, T, dir=dir) + SparseMatrixCSC(T.(adj))


"""
    Dirac(x)

A *Dirac distribution* is parameterized by its only value `x`, and takes its value with probability 1.

```math
P(X = \\hat{x}) = \\begin{cases}
1 & \\quad \\text{for } \\hat{x} = x, \\\\
0 & \\quad \\text{for } \\hat{x} \\neq x.
\\end{cases}
```

```julia
Dirac(2.5)   # Dirac distribution with value x = 2.5
```

External links:

* [Dirac measure on Wikipedia](http://en.wikipedia.org/wiki/Dirac_measure)
"""
struct Dirac{T} <: DiscreteUnivariateDistribution
    value::T
end

Base.eltype(::Type{Dirac{T}}) where {T} = T

insupport(d::Dirac, x::Real) = x == d.value
minimum(d::Dirac) = d.value
maximum(d::Dirac) = d.value
support(d::Dirac) = (d.value,)

#### Properties
mean(d::Dirac) = d.value
var(d::Dirac{T}) where {T} = zero(T)

mode(d::Dirac) = d.value

entropy(d::Dirac{T}) where {T} = zero(T)

#### Evaluation

pdf(d::Dirac, x::Real) = insupport(d, x) ? 1.0 : 0.0
logpdf(d::Dirac, x::Real) = insupport(d, x) ? 0.0 : -Inf

cdf(d::Dirac, x::Real) = x < d.value ? 0.0 : 1.0
cdf(d::Dirac, x::Integer) = x < d.value ? 0.0 : 1.0

quantile(d::Dirac{T}, p::Real) where {T} = 0 <= p <= 1 ? d.value : T(NaN)

mgf(d::Dirac, t) = exp(t * d.value)
cf(d::Dirac, t) = cis(t * d.value)

#### Sampling

rand(rng::AbstractRNG, d::Dirac) = d.value


using Revise
	using SparseArrays, LinearAlgebra, DiffEqOperators, Setfield, Parameters
	using PseudoArcLengthContinuation
	using Plots
	const PALC = PseudoArcLengthContinuation
################################################################################
# case of the SH equation
norminf(x) = norm(x, Inf64)
Nx = 200; Lx = 30.;
X = -Lx .+ 2Lx/Nx*(0:Nx-1) |> collect
hx = X[2]-X[1]

Q = Neumann0BC(hx)
# Q = Dirichlet0BC(hx |> typeof)
Dxx = sparse(CenteredDifference(2, 2, hx, Nx) * Q)[1]
Lsh = -(I + Dxx)^2

function R_SH(u, par)
	@unpack p, b, L1 = par
	out = similar(u)
	out .= L1 * u .- p .* u .+ b .* u.^3 - u.^5
end

Jac_sp = (u, par) -> par.L1 + spdiagm(0 => -par.p .+ 3*par.b .* u.^2 .- 5 .* u.^4)

sol0 = 1.65cos.(X) .* exp.(-X.^2/(2*5^2))
	parSH = (p = 0.7, b = 2., L1 = Lsh)
	optnew = PALC.NewtonPar(verbose = true, tol = 1e-12)
	# allocations 26.47k, 0.038s, tol = 1e-10
	sol1, hist, flag = @time PALC.newton(
		R_SH, Jac_sp,
		sol0, parSH, optnew, normN = norminf)
	Plots.plot(X, sol1)

opts = PALC.ContinuationPar(dsmin = 0.0005, dsmax = 0.0055, ds = -0.001,
		newtonOptions = optnew,
		maxSteps = 1200,
		theta = .6, plotEveryNsteps = 200, computeEigenValues = true)
	br, u1 = @time PALC.continuation(R_SH, Jac_sp, sol1, parSH, (@lens _.p), opts,
		verbosity = 2,
		plot = true,
		# tangentAlgo = BorderedPred(),
		linearAlgo  = MatrixBLS(),
		plotSolution = (x, p;kwargs...)->(plot!(X, x; ylabel="solution", label="", kwargs...)), normC = norminf)
	brs = [br]
#####################################################
# case with computation of eigenvalues
# optnew = PALC.NewtonPar(linsolver = Default(),	eigsolver = eig_KrylovKit{Float64}())
plot(brs, label = "")

####################################################################################################
sol0 = 1.1cos.(X) .* exp.(-0X.^2/(2*5^2))
	optnew = PALC.NewtonPar(verbose = true, tol = 1e-12)
	# allocations 26.47k, 0.038s, tol = 1e-10
	sol1, hist, flag = @time PALC.newton(
	R_SH, Jac_sp,
	sol0, (@set parSH.p = -1.95), optnew)
	Plots.plot(X, sol1)

opts = PALC.ContinuationPar(dsmin = 0.001, dsmax = 0.005, ds = 0.001,
		newtonOptions = setproperties(optnew; maxIter = 30, tol = 1e-11), pMin = -2.,
		maxSteps = 1000, theta = .4, plotEveryNsteps = 200, computeEigenValues = true)
	@assert opts.a<=1.5 "sinon ca peut changer le sens du time step"

	br, u1 = @time PALC.continuation(
		R_SH, Jac_sp, sol1, (@set parSH.p = -1.), (@lens _.p), opts,
		verbosity = 2,
		plot = true,
		# tangentAlgo = BorderedPred(),
		# linearAlgo  = MatrixBLS(),
		plotSolution = (x, p;kwargs...)->(plot!(X, x; ylabel="solution", label="", kwargs...)))

push!(brs, br)

plot(brs)


uniform(u::Real) = 1//2 * float64(abs(u) <= 1.0)
uniform_unnormalized(u::Real) = 1//2 * float64(abs(u) <= 1.0)
Base.@vectorize_1arg Real uniform
Base.@vectorize_1arg Real uniform_unnormalized

triangular(u::Real) = (1 - abs(u)) * float64(abs(u) <= 1.0)
triangular_unnormalized(u::Real) = (1 - abs(u)) * float64(abs(u) <= 1.0)
Base.@vectorize_1arg Real triangular
Base.@vectorize_1arg Real triangular_unnormalized

epanechnikov(u::Real) = 3//4 * (1 - u^2) * float64(abs(u) <= 1.0)
epanechnikov_unnormalized(u::Real) = (1 - u^2) * float64(abs(u) <= 1.0)
Base.@vectorize_1arg Real epanechnikov
Base.@vectorize_1arg Real epanechnikov_unnormalized

biweight(u::Real) = 15//16 * (1 - u^2)^2 * float64(abs(u) <= 1.0)
biweight_unnormalized(u::Real) = (1 - u^2)^2 * float64(abs(u) <= 1.0)
Base.@vectorize_1arg Real biweight
Base.@vectorize_1arg Real biweight_unnormalized

triweight(u::Real) = 35//32 * (1 - u^2)^3 * float64(abs(u) <= 1.0)
triweight_unnormalized(u::Real) = (1 - u^2)^3 * float64(abs(u) <= 1.0)
Base.@vectorize_1arg Real triweight
Base.@vectorize_1arg Real triweight_unnormalized

tricube(u::Real) = 70//81 * (1 - abs(u)^3)^3 * float64(abs(u) <= 1.0)
tricube_unnormalized(u::Real) = (1 - abs(u)^3)^3 * float64(abs(u) <= 1.0)
Base.@vectorize_1arg Real tricube
Base.@vectorize_1arg Real tricube_unnormalized

gaussian(u::Real) = (1 / sqrt(2 * pi)) *  exp(-1//2 * u^2)
gaussian_unnormalized(u::Real) = exp(-1//2 * u^2)
Base.@vectorize_1arg Real gaussian
Base.@vectorize_1arg Real gaussian_unnormalized

cosine(u::Real) = (pi / 4) * cos((pi / 2) * u) * float64(abs(u) <= 1.0)
cosine_unnormalized(u::Real) = cos((pi / 2) * u) * float64(abs(u) <= 1.0)
Base.@vectorize_1arg Real cosine
Base.@vectorize_1arg Real cosine_unnormalized

logistic(u::Real) = 1 / (exp(u) + 2 + exp(-u))
logistic_unnormalized(u::Real) = 1 / (exp(u) + 2 + exp(-u))
Base.@vectorize_1arg Real logistic
Base.@vectorize_1arg Real logistic_unnormalized


using MAT, Plots

@views function generate_grf3D(lx=100.0, ly=100.0, lz=100.0, sf=1.0, cl=(10.0, 8.0, 5.0), k_m=100.0, nh=10000, nx=64, ny=64, nz=64;
                               cov_typ="expon", do_reset=true, do_viz=true, do_save=false)
    # Derived numerics
    dx, dy, dz  = lx/nx, ly/ny, lz/nz  # numerical grid step size
    me = 0
    co1, co2, co3 = 0, 0, 0
    # Array allocation
    Yf      = @zeros(nx, ny, nz)
    # Visu init
    if do_viz
        y_sl    = Int(ceil(ny/2))
        X, Y, Z = -lx/2:dx:lx/2, -ly/2:dy:ly/2, -lz/2:dz:lz/2
    end

    if cov_typ=="expon"
        # Generate the 3D exponential covariance function
        wtime_it = grf3D_expon!(Yf, sf, cl, nh, nx, ny, nz, dx, dy, dz; me, co1, co2, co3, do_reset)
    elseif cov_typ=="gauss"
        # Generate the 3D Gaussian covariance function
        wtime_it = grf3D_gauss!(Yf, sf, cl[1]/2.0, nh, k_m, nx, ny, nz, dx, dy, dz; me, co1, co2, co3, do_reset)
    else
        error("Undefined covariance function.")
    end

    # Performance
    A_eff    = 2/1e9*nx*ny*nz*sizeof(Data.Number)  # Effective main memory access per iteration [GB] (Lower bound of required memory access: H and dHdτ have to be read and written (dHdτ for damping): 4 whole-array memaccess; B has to be read: 1 whole-array memaccess)
    T_eff    = A_eff/wtime_it                      # Effective memory throughput [GB/s]
    if (me==0) @printf("T_eff = %1.2f GB/s \n", round(T_eff, sigdigits=2)) end

    # Visualisation
    if do_viz
        display(heatmap(X, Z, Array(Yf)[:,y_sl,:]', aspect_ratio=1, xlims=(X[1],X[end]), ylims=(Z[1],Z[end]), c=:hot, title="3D RandomField (y-slice)"))
    end
    if do_save  file = matopen("grf3D_$(cov_typ).mat", "w"); write(file, "grf3D", Array(Yf)); close(file)  end

    return Yf
end


using GNSSTools
using JLD
using PyPlot
pygui(true)

GM = 3.986004418e14  # m³s⁻²
eop = get_eop();

# N_rand = 25
# lats = rand(-90:0.001:90, N_rand)  # degrees
# longs = rand(-180:0.001:180, N_rand)  # degrees
# heights = fill(100, N_rand)  # meters
# llas = (lats, longs, heights)
# user_lla = llas
freq = L1_freq
# user_lla = (40.01, -105.2437, 1655);

file_name = "/home/sjbilardi/Dropbox/Apps/Overleaf/MastersThesis/ch1_doppler_curves_user_lla.jld"
# file_name = "ch1_doppler_curves_user_lla.jld"
user_lla = load(file_name, "user_lla")

Δt = 1  # seconds
total_time = 1*24*60*60  # seconds
t_range = Array(0:Δt:total_time);
min_elevation = 10  # degrees

# GPS
a = 26_560e3;  # m
plane_num = 6;
sat_per_plane = 4;
incl = 56;
ΔΩ = 360/plane_num;
Δf_per_plane = 360/sat_per_plane/2;  # degrees
orbital_period = 2π*sqrt(a^3 / GM)  # seconds
doppler_gps, doppler_rate_gps, doppler_bounds_gps,
doppler_rate_bounds_gps, raw_dopplers_gps = doppler_distribution(a, plane_num,
                                           sat_per_plane, incl,
                                           t_range, user_lla,
                                           freq, show_plot=false,
                                           print_steps=true,
                                           eop=eop,
                                           min_elevation=min_elevation,
                                           ΔΩ=ΔΩ,
                                           Δf_per_plane=Δf_per_plane,
										   return_constellation=false,
                                           return_raw_dopplers=true);
doppler_raw_gps, doppler_rate_raw_gps,
elevations_gps, max_elevations_gps = raw_dopplers_gps

# Iridium
a = Rₑ+780*1000;  # meters
plane_num = 6;
sat_per_plane = 11;
incl = 86;
ΔΩ = 360/plane_num/2;
Δf_per_plane = 360/sat_per_plane/2;  # degrees
orbital_period = 2π*sqrt(a^3 / GM)  # seconds
doppler_iridium, doppler_rate_iridium, doppler_bounds_iridium,
doppler_rate_bounds_iridium, raw_dopplers_iridium = doppler_distribution(a, plane_num,
                                            sat_per_plane, incl,
                                            t_range, user_lla,
                                            freq, show_plot=false,
                                            print_steps=true,
                                            eop=eop,
                                            ΔΩ=ΔΩ,
                                            Δf_per_plane=Δf_per_plane,
                                            return_raw_dopplers=true);
doppler_raw_iridium, doppler_rate_raw_iridium,
elevations_iridium, max_elevations_iridium = raw_dopplers_iridium

# Starlink
a = Rₑ+550*1000;  # meters
plane_num = 24;
sat_per_plane = 66;
incl = 53;
ΔΩ = 360/plane_num;
Δf_per_plane = 360/sat_per_plane/2;  # degrees
orbital_period = 2π*sqrt(a^3 / GM)  # seconds
doppler_starlink, doppler_rate_starlink, doppler_bounds_starlink,
doppler_rate_bounds_starlink, raw_dopplers_starlink = doppler_distribution(a, plane_num,
                                            sat_per_plane, incl,
                                            t_range, user_lla,
                                            freq, show_plot=false,
                                            print_steps=true,
                                            eop=eop,
                                            ΔΩ=ΔΩ,
                                            Δf_per_plane=Δf_per_plane,
                                            return_raw_dopplers=true);
doppler_raw_starlink, doppler_rate_raw_starlink,
elevations_starlink, max_elevations_starlink = raw_dopplers_starlink

# OneWeb
a = Rₑ+1200*1000;  # meters
plane_num = 18;
sat_per_plane = 36;
incl = 87.9;  # degrees
ΔΩ = 360/plane_num/2;
Δf_per_plane = 360/sat_per_plane/2;  # degrees
orbital_period = 2π*sqrt(a^3 / GM)  # seconds
doppler_oneweb, doppler_rate_oneweb, doppler_bounds_oneweb,
doppler_rate_bounds_oneweb, raw_dopplers_oneweb = doppler_distribution(a, plane_num,
                                            sat_per_plane, incl,
                                            t_range, user_lla,
                                            freq, show_plot=false,
                                            print_steps=true,
                                            eop=eop,
                                            ΔΩ=ΔΩ,
                                            Δf_per_plane=Δf_per_plane,
                                            return_raw_dopplers=true);
doppler_raw_oneweb, doppler_rate_raw_oneweb,
elevations_oneweb, max_elevations_oneweb = raw_dopplers_oneweb

save("ch1_doppler_curves.jld",
     "doppler_gps", doppler_gps,
	 "doppler_iridium", doppler_iridium,
	 "doppler_starlink", doppler_starlink,
	 "doppler_oneweb", doppler_oneweb,
	 "doppler_rate_gps", doppler_rate_gps,
	 "doppler_rate_iridium", doppler_rate_iridium,
	 "doppler_rate_starlink", doppler_rate_starlink,
	 "doppler_rate_oneweb", doppler_rate_oneweb,
	 "doppler_bounds_gps", doppler_bounds_gps,
	 "doppler_bounds_iridium", doppler_bounds_iridium,
	 "doppler_bounds_starlink", doppler_bounds_starlink,
	 "doppler_bounds_oneweb", doppler_bounds_oneweb,
	 "doppler_raw_gps", doppler_raw_gps,
	 "doppler_raw_iridium", doppler_raw_iridium,
	 "doppler_raw_starlink", doppler_raw_starlink,
	 "doppler_raw_oneweb", doppler_raw_oneweb,
	 "doppler_rate_raw_gps", doppler_rate_raw_gps,
	 "doppler_rate_raw_iridium", doppler_rate_raw_iridium,
	 "doppler_rate_raw_starlink", doppler_rate_raw_starlink,
	 "doppler_rate_raw_oneweb", doppler_rate_raw_oneweb,
	 "freq", freq,
	 "user_lla", user_lla,
	 "t_range", t_range,
	 "min_elevation", min_elevation,
	 "elevations_gps", elevations_gps,
	 "elevations_iridium", elevations_iridium,
	 "elevations_starlink", elevations_starlink,
	 "elevations_oneweb", elevations_oneweb,
	 "max_elevations_gps", max_elevations_gps,
	 "max_elevations_iridium", max_elevations_iridium,
	 "max_elevations_starlink", max_elevations_starlink,
	 "max_elevations_oneweb", max_elevations_oneweb)

# file_name = "/home/sjbilardi/Dropbox/Apps/Overleaf/MastersThesis/ch1_doppler_curves.jld"
# data = load(file_name,
#             "doppler_gps",
# 	       "doppler_iridium",
# 	       "doppler_starlink",
# 	       "doppler_oneweb",
# 	       "doppler_rate_gps",
# 	       "doppler_rate_iridium",
# 	       "doppler_rate_starlink",
# 	       "doppler_rate_oneweb",
# 	       "doppler_bounds_gps",
# 	       "doppler_bounds_iridium",
# 	       "doppler_bounds_starlink",
# 	       "doppler_bounds_oneweb",
# 	       "doppler_raw_gps",
# 	       "doppler_raw_iridium",
# 	       "doppler_raw_starlink",
# 	       "doppler_raw_oneweb",
# 	       "doppler_rate_raw_gps",
# 	       "doppler_rate_raw_iridium",
# 	       "doppler_rate_raw_starlink",
# 	       "doppler_rate_raw_oneweb",
# 	       "freq",
# 	       "user_lla",
# 	       "t_range",
# 	       "min_elevation")

# doppler_gps, doppler_iridium, doppler_starlink, doppler_oneweb,
# doppler_rate_gps, doppler_rate_iridium, doppler_rate_starlink, doppler_rate_oneweb,
# doppler_bounds_gps, doppler_bounds_iridium, doppler_bounds_starlink, doppler_bounds_oneweb,
# doppler_raw_gps, doppler_raw_iridium, doppler_raw_starlink, doppler_raw_oneweb,
# doppler_rate_raw_gps, doppler_rate_raw_iridium, doppler_rate_raw_starlink, doppler_rate_raw_oneweb,
# freq, user_lla, t_range, min_elevation = data

# # Make plots
# fig = figure(figsize=(8.5,9))
# # GPS
# ax1 = fig.add_subplot(4,2,1)
# ax1.hist(Array(doppler_gps.edges[1])[2:end]./1000,
#          Array(doppler_gps.edges[1])./1000, weights=doppler_gps.weights,
#          color="k");
# xlabel("Doppler (kHz)")
# ylabel("Probability")
# title("GPS")
# ax2 = fig.add_subplot(4,2,2)
# ax2.hist(Array(doppler_rate_gps.edges[1])[2:end],
#          Array(doppler_rate_gps.edges[1]), weights=doppler_rate_gps.weights,
#          color="k");
# xlabel("Doppler Rate (Hz/s)")
# ylabel("Probability")
# title("GPS")
# # Iridium
# ax3 = fig.add_subplot(4,2,3)
# ax3.hist(Array(doppler_iridium.edges[1])[2:end]./1000,
#          Array(doppler_iridium.edges[1])./1000, weights=doppler_iridium.weights,
#          color="k");
# xlabel("Doppler (kHz)")
# ylabel("Probability")
# title("Iridium")
# ax4 = fig.add_subplot(4,2,4)
# ax4.hist(Array(doppler_rate_iridium.edges[1])[2:end],
#          Array(doppler_rate_iridium.edges[1]), weights=doppler_rate_iridium.weights,
#          color="k");
# xlabel("Doppler Rate (Hz/s)")
# ylabel("Probability")
# title("Iridium")
# # Starlink
# ax5 = fig.add_subplot(4,2,5)
# ax5.hist(Array(doppler_starlink.edges[1])[2:end]./1000,
#          Array(doppler_starlink.edges[1])./1000, weights=doppler_starlink.weights,
#          color="k");
# xlabel("Doppler (kHz)")
# ylabel("Probability")
# title("Starlink")
# ax6 = fig.add_subplot(4,2,6)
# ax6.hist(Array(doppler_rate_starlink.edges[1])[2:end],
#          Array(doppler_rate_starlink.edges[1]), weights=doppler_rate_starlink.weights,
#          color="k");
# xlabel("Doppler Rate (Hz/s)")
# ylabel("Probability")
# title("Starlink")
# # OneWeb
# ax7 = fig.add_subplot(4,2,7)
# ax7.hist(Array(doppler_oneweb.edges[1])[2:end]./1000,
#          Array(doppler_oneweb.edges[1])./1000, weights=doppler_oneweb.weights,
#          color="k");
# xlabel("Doppler (kHz)")
# ylabel("Probability")
# title("OneWeb")
# ax8 = fig.add_subplot(4,2,8)
# ax8.hist(Array(doppler_rate_oneweb.edges[1])[2:end],
#          Array(doppler_rate_oneweb.edges[1]), weights=doppler_rate_oneweb.weights,
#          color="k");
# xlabel("Doppler Rate (Hz/s)")
# ylabel("Probability")
# title("OneWeb")

# subplots_adjust(hspace=0.7, wspace=0.35, top=0.93, left=0.1, right=0.9)
# savefig("/home/sjbilardi/Dropbox/Apps/Overleaf/MastersThesis/figures/ch1_doppler_distributions.svg", dpi=300)

# using PyCall
# basemap = pyimport("mpl_toolkits.basemap")
# # lon_0 is central longitude of projection.
# # resolution = 'c' means use crude resolution coastlines.
# fig = figure(figsize=(7.5, 2.5))
# ax1 = fig.add_subplot(1,1,1)
# m = basemap.Basemap(projection="robin", lon_0=0, resolution="c")
# m.drawcoastlines(color="black", linewidth=0.25)
# m.fillcontinents(color="#69b2a2",lake_color="#A6CAE0")
# # draw parallels and meridians.
# m.drawparallels(range(-90., 120., step=30.), linewidth=0.5, color="dimgrey")
# m.drawmeridians(range(0., 360., step=60.), linewidth=0.5, color="dimgrey")
# m.drawmapboundary(fill_color="#A6CAE0")
# # m.fillcontinents(color="grey", alpha=0.3)
# color = "tomato"
# marker = "."
# markersize = 5
# N = min(length(user_lla[1]), length(user_lla[2]), length(user_lla[3]))
# for i in 1:N
# 	lat = user_lla[1][i]
# 	lon	= user_lla[2][i]
# 	xpt, ypt = m(lon,lat)
# 	m.plot(xpt, ypt, color=color, marker=marker, markersize=markersize)
# end
# # title("$N Random Observation Locations")
# subplots_adjust(bottom=0.01, left=0.1, right=0.9, top=0.99)
# savefig("/home/sjbilardi/Dropbox/Apps/Overleaf/MastersThesis/figures/ch1_doppler_curves_random_location_map.pdf",
#         dpi=300)


using SpecialFunctions

"""
    G(x::Real)

Computes the second synchrotron function at a given frequency ratio ``x = \\frac{\\nu}{\\nu_0}``.

``G(x) = x K_{\\frac{2}{3}}(x)``
"""
G(x::Real) = x * besselk(2/3, x)

"""
    𝒢(x::Real)

Computes the second synchrotron function at a given frequency ratio ``x = \\frac{\\nu}{\\nu_0}``.

``G(x) = x K_{\\frac{2}{3}}(x)``

Wrapper for [`G`](@ref).
"""
𝒢(x::Real) = G(x)


function sphcesaro1(A::AbstractMatrix{T}) where T
    M, N = size(A)
    m, n = M, N÷2
    C = zeros(T, m, 2n+1)
    @inbounds for i = 1:m
        C[i,1] = (m+one(T)-i)/m
    end
    @inbounds for j = 1:n
        for i = 1:m-j
            C[i,2j] = C[i+j,1]
            C[i,2j+1] = C[i+j,1]
        end
    end
    C
end

function sphcesaro2(A::AbstractMatrix{T}) where T
    M, N = size(A)
    m, n = M, N÷2
    C = zeros(T, m, 2n+1)
    @inbounds for i = 1:m
        C[i,1] = (m+2one(T)-i)*(m+one(T)-i)/((m+one(T))*m)
    end
    @inbounds for j = 1:n
        for i = 1:m-j
            C[i,2j] = C[i+j,1]
            C[i,2j+1] = C[i+j,1]
        end
    end
    C
end

function sphcesaro(A::AbstractMatrix{T}, κ::Int) where T
    M, N = size(A)
    m, n = M, N÷2
    C = zeros(T, m, 2n+1)
    @inbounds for i = 1:m
        C[i,1] = binomial(m+κ-i, m-i)/binomial(m+κ-1, m-1)
    end
    @inbounds for j = 1:n
        for i = 1:m-j
            C[i,2j] = C[i+j,1]
            C[i,2j+1] = C[i+j,1]
        end
    end
    C
end